As is known, on a factory floor of a food packaging plant, several specifically-aimed processes are generally performed, including incoming food and packaging material storage, food processing, food packaging, and package warehousing. With specific reference to pourable food products, food packaging is performed in Packaging Lines, each of which is an assembly of machines and equipments for the production and handling of packages, and includes a Filling Machine for the production of sealed packages, followed by one or more defined configurations of downstream Distribution Equipments such as, accumulators, straw applicators, film wrappers, and cardboard packers, connected to the Filling Machine via Conveyors, for the handling of the packages.
A typical example of this type of packages is the parallelepiped-shaped package for liquid or pourable food products known as Tetra Brik Aseptic®, which is made by folding and sealing a laminated web of packaging material.
The packaging material has a multilayer sheet structure substantially comprising one or more stiffening and strengthening base layers typically made of a fibrous material, e.g. paper, or mineral-filled polypropylene material, covered on both sides with a number of heat-seal plastic material layers, e.g. polyethylene film. In the case of aseptic packages for long-storage products, such as UHT milk, the packaging material also comprises a gas- and light-barrier material layer, e.g. aluminium foil or ethyl vinyl alcohol (EVOH) film, which is superimposed on a heat-seal plastic material layer, and is in turn covered with another heat-seal plastic material layer forming the inner face of the package eventually contacting the food product.
Packages of this sort are produced on fully automatic Filling Machines, wherein a continuous vertical tube is formed from the web-fed packaging material; which is sterilized by applying a chemical sterilizing agent such as a hydrogen peroxide solution, which, once sterilization is completed, is removed, e.g. evaporated by heating, from the surfaces of the packaging material; and the sterilized web is maintained in a closed, sterile environment, and is folded and sealed longitudinally to form the vertical tube. The tube is then filled downwards with the sterilized or sterile-processed pourable food product, and is fed along a vertical path to a forming station, where it is gripped along equally spaced cross sections by a jaw system including two or more pairs of jaws, which act cyclically and successively on the tube, and seal the packaging material of tube to form a continuous strip of pillow packs connected to one another by transverse sealing strips. Pillow packs are separated from one another by cutting the relative sealing strips, and are conveyed to a final folding station where they are folded mechanically into the finished, e.g. substantially parallelepiped-shaped, packages.
Alternatively, the packaging material may be cut into blanks, which are formed into packages on forming spindles, and the packages are filled with food product and sealed. One example of this type of package is the so-called “gable-top” package known as Tetra Rex®.
In these Packaging Lines, several components are operated by (electric) servomotors, which, although valuable in several respects, are affected by malfunctions, one of the major causes of which is the breakdown of the rolling bearings supporting the shaft of the servomotors due to fatigue or wear. While fatigue can be statistically characterized in a standard way that leads to the so-called L10 rating of the bearings, wear is a subtler phenomenon, known in the literature as pitting or brinelling, that creates localized damages, the onset of which may (pseudo)randomly appear during the expected lifetime of the component, followed by a relatively quick degradation phase that leads to the complete breakdown. As a result, periodic replacement of these components is a strategy that may be successful only to prevent fatigue-related failures, whereas it will be almost useless against wear-related failures.
Other kinds of preventive periodic maintenance activities, such as cleaning and lubrication, may be effective in reducing bearing wear, as usually wear is characterized by the contamination of the bearing lubricant which in turn worsens the bearing health. Such contamination may come from the exterior (e.g. the bearing is placed in a hostile environment) but may also be due to internal causes (e.g. due to small flakes of material that the revolving elements lose during operation).
Ultimately, however, the only way to prevent wear-related breakdown is condition-based monitoring of the bearing health status; such maintenance strategy is mainly possible thanks to the fact that once the bearing approaches failure, it becomes noisy and vibrates as a warning sign of the impending breakdown: if this sign is detected timely it gives the operator a time frame that typically ranges from days to even weeks (depending on the bearing and the application) to plan a maintenance activity and substitute the bearing without impacting production time.
It may be appreciated that vibration analysis is an important part of industrial predictive maintenance programs so that wear and damages in the rolling bearings can be discovered and repaired before the machine breaks down, thus reducing operating and maintenance costs.
Empirical evaluation of the vibration level of a bearing is an error-prone activity that may lead to significantly underestimate or overestimate the remaining lifetime of the component, and also to mistake for a bearing damage a noise that is due to a completely different cause (e.g. a shaft imbalance). For this reason, the scientific community has striven to provide a scientific characterization of bearing faults, and nowadays a rich literature can be found on this topic.
The basic idea is associating each failure mode of a bearing with a characteristic frequency signature, that can be extracted from a vibration signal via an appropriate analysis. In particular, traditional vibration analysis is based on the fact that if there is a localized damage on one of the bearing surfaces, it will cause a series of impacts during bearing rotation; moreover, such impacts are periodic assuming that the servomotor is rotating at constant speed. In fact, a kinematic analysis of the bearing shows that, assuming that no slip is present, the component is quite similar to an epicyclic gear; in other words there is a fixed “transmission ratio” between the servomotor shaft and all the other moving parts of the bearing, and this leads to the fundamental equation of vibration analysis:fd=kdfr  (1)which shows that fd, the damage frequency (which is actually the frequency of such impacts) depends linearly on the rotation frequency fr by means of a damage coefficient kd which is none other than the transmission ratio between the servomotor shaft and the moving part on which the damage is located. Such coefficients are well-known in the literature and are given by:
                              k          d                =                  {                                                                                                                1                      2                                        ⁢                                          (                                              1                        -                                                                                                            B                              d                                                                                      P                              d                                                                                ⁢                          cos                          ⁢                                                                                                          ⁢                          θ                                                                    )                                                        =                                      k                    g                                                                                                for                  ⁢                                                                          ⁢                  cage                  ⁢                                                                          ⁢                  faults                                                                                                                                                1                      2                                        ⁢                                          N                      ⁡                                              (                                                  1                          -                                                                                                                    B                                d                                                                                            P                                d                                                                                      ⁢                            cos                            ⁢                                                                                                                  ⁢                            θ                                                                          )                                                                              =                                      k                    e                                                                                                for                  ⁢                                                                          ⁢                  outer                  ⁢                                                                          ⁢                  ring                  ⁢                                                                          ⁢                  faults                                                                                                                                                1                      2                                        ⁢                                          N                      ⁡                                              (                                                  1                          +                                                                                                                    B                                d                                                                                            P                                d                                                                                      ⁢                            cos                            ⁢                                                                                                                  ⁢                            θ                                                                          )                                                                              =                                      k                    i                                                                                                for                  ⁢                                                                          ⁢                  inner                  ⁢                                                                          ⁢                  ring                  ⁢                                                                          ⁢                  faults                                                                                                                                                1                      2                                        ⁢                                                                  B                        d                                                                    P                        d                                                              ⁢                                          N                      ⁡                                              [                                                  1                          -                                                                                    (                                                                                                                                    B                                    d                                                                                                        P                                    d                                                                                                  ⁢                                cos                                ⁢                                                                                                                                  ⁢                                θ                                                            )                                                        2                                                                          ]                                                                              =                                      k                    v                                                                                                for                  ⁢                                                                          ⁢                  ball                  ⁢                                                                          ⁢                  faults                                                                                        (        2        )            where Bd, Pd are ball and pitch diameter, N is the number of revolving elements, θ is an angle that indicates a possible misalignment between inner and outer rings (or rings) as a result of mounting operations (typical values are between 0 and 10 degrees), and wherein the subscript d relates in general to damage frequencies, while subscripts g, e, i and v relate to specific kinds of damage, namely in the inner and outer rings, in the balls evenly angularly spaced apart between the inner and outer rings, and in the cage which retains the balls and which rotates jointly, i.e., at the same speed, with the balls.
The repercussions of this phenomenon on the spectrum of a vibration signal can be easily understood using the basic properties of Fourier transforms: if a single impact is considered, in the time domain this can be represented as a forcing action d(t) of impulsive nature and finite duration T, which tends to an ideal Dirac impulse function δ(t) as T→0; likewise, the spectrum D(f) of said signal will be characterized by a bandwidth that will tend to infinity as d(t) approaches the ideal impulse case (whose Fourier transform is constant over all frequencies).
During the actual component operation the forcing action u(t) will be a periodic repetition at the damage frequency of the original impulse d(t):
                              u          ⁡                      (            t            )                          =                              ∑                          j              =                              -                ∞                                      ∞                    ⁢                      d            ⁡                          (                              t                -                                  j                  ⁢                                                                          ⁢                                      T                    d                                                              )                                                          (        3        )            
By virtue of the properties of the Fourier transform, periodic repetition in time equates to sampling in frequency, which means that the spectrum U(f) of the forcing action that a localized damage applies to the bearing is a discrete spectrum, obtained by sampling the original impulse spectrum D(f):
                              U          ⁡                      (            f            )                          =                              ∑                          j              =                              -                ∞                                      ∞                    ⁢                                    D              ⁡                              (                f                )                                      ⁢                          δ              ⁡                              (                                  f                  -                                      j                    ⁢                                                                                  ⁢                                          f                      d                                                                      )                                                                        (        4        )            
In practice, this means that the frequency signature of the damage in the vibration signal is a sequence of peaks separated by the characteristic damage frequency of the bearing part damaged.
In general, however, the spectrum of a vibration signal acquired on a bearing does not reproduce exactly U(f); it can be rather represented as:Y(f)=G(f)H(f)U(f)+N(f)  (5)where G(f) is the transfer function of the mechanical assembly, H(f) is the sensor (usually an accelerometer) sensitivity function, and N(f) is any kind of noise superimposed to the fault signal. The spectrum of Y(f) as such may not therefore be the best signal to look at in order to identify a damage signature; the usual procedure to obtain a signal with better signal-to-noiseratio is called envelope analysis and is based on the following assumptions: there exist a frequency band [f1, f2] such that:                |G(f)|>>1, that is we are near a mechanical resonance        |H(f)|>>1, that is we are in the operational range of the sensor        |N(f)|<<1, which means in practice that we have to look for higher harmonics of the signal U(f).        
The latter statement is motivated by the fact that usually mechanical noise is higher at low frequencies. There are some exceptions to this rule, for example noise due to gearing attached to the motor, and in this case it is necessary to resort to more advanced known filtering techniques. If the above hypotheses are satisfied, by band-pass filtering in the frequency band [f1, f2] and demodulation of Y(f) it is possible to obtain a signal where some peaks are clearly visible, spaced by fd.
Summing up, classical vibration analysis is based upon the following hypotheses:                there is a localized damage on a bearing;        the motor the bearing is attached to rotates at a constant speed;        there is no slip during the relative motion of the bearing elements;        during the motor operation, the damage causes a series of short-duration impacts, that generate a train of spikes in the frequency spectrum of the vibration signal with a certain periodicity; and        there is a frequency band where the signal-to-noise ratio is such that the train of impulses is detectable.        
If these conditions are not verified, the train of peaks may be smeared so that it is not recognizable anymore, or can be hidden among other kinds of noise. Moreover, the fact that the angle θ in (2) is almost impossible to measure under practical circumstances adds some difficulties to the task, as each fd is actually variable in the range of admissible θ's. Most research in the field has historically been focused to signal processing techniques to obtain better signal-to-noise ratios or to cope with smearing of the peaks due to small fluctuations of velocity or the presence of slip.
Despite the basic assumption of constant rotation speed of the servomotors still holds true for many applications, it proves to be a huge limitation in the field of automatic machines, where usually a number of servomotors are employed as electric cams and operated at a variable speed in order to obtain variable speed profiles of the actuated elements. In particular, as servomotors, usually AC brushless motors, tend to appear more and more often in recent machine designs thanks to their performance being much higher than the mechanical solutions for machine motion in time required to reconfigure the motion profile, in these applications any rolling bearing fault prediction based on frequency signature determined via the above-described classical vibration analysis proves to be unsatisfactory.
In order to extend the above-described classical vibration analysis based on the constant rotation speed assumption to cope with applications in which the rotation speed of the servomotors varies over time, a so-called Order Tracking (OT) vibration analysis has been proposed, which is a frequency analysis that uses multiples, commonly referred to as orders, of the rotation speed, instead of absolute frequencies (Hz) as the frequency base, and is useful for machine condition monitoring because it can easily identify speed-related vibrations such as shaft defects and bearing wear. For a detailed discussion of this technique, reference may be made to R. Potter, A new order tracking method for rotating machinery, Sound and Vibration 24, 1990, 30-34, and K. Fyfe, E. Munck, Analysis of computed order tracking, Mechanical System and Signal Processing 11(2), 1997, 187-205.
Order Tracking is based on a constant spatial sampling approach, according to which the vibration signal is sampled at constant angular increments (i.e. uniform Δθ), and hence at a frequency proportional to the bearing rotation speed. Traditional order tracking accomplishes this task by using a variable-time data acquisition system, wherein a time-sampling is performed at a frequency which is varied proportionally to the bearing rotation speed. Computed order tracking (COT), instead, accomplishes this task by using a constant-time data acquisition system, wherein the vibration signal is first time-sampled at a constant frequency (i.e. constant-time sampling with uniform Δt), and then the sampled data are digitally spatially re-sampled at constant angular increments (i.e. constant-space sampling with uniform Δθ) to provide the desired constant Δθ data. Therefore, Order Tracking maps real time reference t to a modified time reference τ such that the vibration signal is seen as if it were generated piece-wise by a bearing rotating at a constant speed. This mapping results in a modified vibration signal where the frequency signature of a given typology of damage can be found, and in a rolling bearing fault being satisfactorily predicted.